Differentiation: Quotient rule
The quotient rule
We saw earlier that we can multiply and compose functions. We can also divide functions. We call the result the quotient. The quotient of the functions #\blue{g(x)}=\blue{x^3+1}# and #\green{h(x)}=\green{x+1}# is the function #f(x)=\frac{\blue{x^3+1}}{\green{x+1}}#. In this case, same as with regular fractions, we call #\blue{g(x)}# the numerator and #\green{h(x)}# the denominator.
We can calculate the derivative of a quotient by means of the quotient rule.
Quotient rule
For the quotient of two functions \[f(x)=\dfrac{\blue{g(x)}}{\green{h(x)}}\] the following applies:
\[f'(x)=\dfrac{\green{h(x)}\cdot \orange{g'(x)}\blue{g(x)}\cdot \purple{h'(x)}}{(\green{h(x)})^2}\]
Example
\[f(x)=\dfrac{\blue{2x}}{\green{x+1}}\] gives \[f'(x)=\dfrac{(\green{x+1})\cdot \orange{2}\blue{2x}\cdot \purple{1}}{(\green{x+1})^2
}\]
We can follow the stepbystep guide below to apply the quotient rule.
Stepbystep guide quotient rule
Stepbystep 
Example 

Consider the #f(x)#, which is a quotient of two functions. 
#\dfrac{\sin(x)}{x^2+1}# 

Step 1 
Distinguish the numerator #\blue{g(x)}# and the denominator #\green{h(x)}#. 
#\blue{g(x)}=\blue{\sin(x)}# #\green{h(x)}=\green{x^2+1}# 
Step 2 
Calculate #\orange{g'(x)}# and #\purple{h'(x)}#. 
#\orange{g'(x)}=\orange{\cos(x)}# #\purple{h'(x)}=\purple{2x}# 
Step 3 
Calculate the derivative of #f# using the formula: \[f'(x)=\dfrac{\green{h(x)}\cdot \orange{g'(x)}\blue{g(x)}\cdot \purple{h'(x)}}{(\green{h(x)})^2}\] 
#\dfrac{(\green{x^2+1})\cdot \orange{\cos(x)}\blue{\sin(x)}\cdot \purple{2x}}{(\green{x^2+1})^2}# 
Step 1  We determine #g(x)# and #h(x)# such that #f(x)=\frac{g(x)}{h(x)}#. #\begin{array}{rcl} g(x)&=&x+5\\ h(x)&=&4\cdot x^4+5\end{array}# 
Step 2  We calculate the derivative #g'(x)# and #h'(x)#. #\begin{array}{rcl} g'(x)&=&\dfrac{\dd}{\dd x}\left(x+5\right)\\ &&\blue{\text{definition derivative}}\\ &=&1\\ &&\blue{\text{sum rule, power rule and constant rule}}\end{array}# #\begin{array}{rcl} h'(x)&=&\dfrac{\dd}{\dd x}\left(4\cdot x^4+5\right)\\ &&\blue{\text{definition derivative}}\\ &=&16\cdot x^3\\ &&\blue{\text{sum rule, power rule and constant rule}}\end{array}# 
Step 3 
#\begin{array}{rcl} 
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